2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:53 PM Page 2.1 Gaussian Beam Optics www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics 2 Gaussian Beam Optics Optical Specifications Gaussian Beam Propagation 2.2 Transformation and Magnification by Simple Lenses 2.6 Real Beam Propagation 2.10 Lens Selection 2.13 Material Properties Optical Coatings Gaussian Beam Optics 2.1 2ch_GuassianBeamOptics_Final.qxd 6/16/2009 2:19 PM Page 2.2 Gaussian Beam Optics www.cvimellesgriot.com Gaussian Beam Propagation Fundamental Optics In most laser applications it is necessary to focus, modify, or shape the laser For virtually all laser cavities, the propagation of an electromagnetic field, beam by using lenses and other optical elements. In general, laser-beam E(0), through one round trip in an optical resonator can be described propagation can be approximated by assuming that the laser beam has an mathematically by a propagation integral, which has the general form ideal Gaussian intensity profile, which corresponds to the theoretical TEM00 ()1 − jkp ()0 mode. Coherent Gaussian beams have peculiar transformation properties E() x,, y= e∫∫ K() x y, x00, y E() x00, y dx 0 dy 0 which require special consideration. In order to select the best optics InputPlane (2.1) for a particular laser application, it is important to understand the basic where K is the propagation constant at the carrier frequency of the opti- properties of Gaussian beams. cal signal, p is the length of one period or round trip, and the integral is over Unfortunately, the output from real-life lasers is not truly Gaussian (although the transverse coordinates at the reference or input plane. The function K the output of a single mode fiber is a very close approximation). To accom- is commonly called the propagation kernel since the field E(1)(x, y), after M2 (0) modate this variance, a quality factor, (called the “M-squared” one propagation step, can be obtained from the initial field E (x0, y0) factor), has been defined to describe the deviation of the laser beam from through the operation of the linear kernel or “propagator” K(x, y, x0, y0). a theoretical Gaussian. For a theoretical Gaussian, M2 = 1; for a real laser Gaussian Beam Optics By setting the condition that the field, after one period, will have exactly beam, M2>1. The M2 factor for helium neon lasers is typically less than the same transverse form, both in phase and profile (amplitude variation 1.1; for ion lasers, the M2 factor typically is between 1.1 and 1.3. Collimated 2 across the field), we get the equation TEM00 diode laser beams usually have an M ranging from 1.1 to 1.7. For high-energy multimode lasers, the M2 factor can be as high as 25 or 30. g Exy(),,≡ KxyxyExydxdy(),,() In all cases, the M2 factor affects the characteristics of a laser beam and nm nm ∫∫ 00nm 00, 00 InputPlane (2.2) cannot be neglected in optical designs. where E represents a set of mathematical eigenmodes, and g a In the following section, Gaussian Beam Propagation, we will treat the nm nm corresponding set of eigenvalues. The eigenmodes are referred to as characteristics of a theoretical Gaussian beam (M2=1); then, in the section transverse cavity modes, and, for stable resonators, are closely approx- Real Beam Propagation we will show how these characteristics change as imated by Hermite-Gaussian functions, denoted by TEM . (Anthony the beam deviates from the theoretical. In all cases, a circularly symmetric nm Siegman, Lasers) wavefront is assumed, as would be the case for a helium neon laser or an argon-ion laser. Diode laser beams are asymmetric and often astigmatic, The lowest order, or “fundamental” transverse mode, TEM00 has a which causes their transformation to be more complex. Optical Specifications Gaussian intensity profile, shown in figure 2.1, which has the form Although in some respects component design and tolerancing for lasers is −+()22 , ∝ kx y more critical than for conventional optical components, the designs often Ixy() e (2.3) tend to be simpler since many of the constraints associated with imaging In this section we will identify the propagation characteristics of this low- systems are not present. For instance, laser beams are nearly always used est-order solution to the propagation equation. In the next section, Real Beam on axis, which eliminates the need to correct asymmetric aberration. Propagation, we will discuss the propagation characteristics of higher-order Chromatic aberrations are of no concern in single-wavelength lasers, although modes, as well as beams that have been distorted by diffraction or various they are critical for some tunable and multiline laser applications. In fact, the anisotropic phenomena. only significant aberration in most single-wavelength applications is primary (third-order) spherical aberration. Scatter from surface defects, inclusions, dust, or damaged coatings is of BEAM WAIST AND DIVERGENCE greater concern in laser-based systems than in incoherent systems. Speckle In order to gain an appreciation of the principles and limitations of Material Properties content arising from surface texture and beam coherence can limit system Gaussian beam optics, it is necessary to understand the nature of the laser performance. output beam. In TEM00 mode, the beam emitted from a laser begins as a Because laser light is generated coherently, it is not subject to some of the perfect plane wave with a Gaussian transverse irradiance profile as shown limitations normally associated with incoherent sources. All parts of the in figure 2.1. The Gaussian shape is truncated at some diameter either by wavefront act as if they originate from the same point; consequently, the the internal dimensions of the laser or by some limiting aperture in the emergent wavefront can be precisely defined. Starting out with a well- optical train. To specify and discuss the propagation characteristics of a defined wavefront permits more precise focusing and control of the beam laser beam, we must define its diameter in some way. There are two than otherwise would be possible. commonly accepted definitions. One definition is the diameter at which the beam irradiance (intensity) has fallen to 1/e2 (13.5 percent) of its peak, or axial value and the other is the diameter at which the beam irradiance Optical Coatings 2.2 Gaussian Beam Optics 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:54 PM Page 2.3 Gaussian Beam Optics www.cvimellesgriot.com Fundamental Optics toward infinity as z is further increased, asymptotically approaching the 100 value of z itself. The plane z = 0 marks the location of a Gaussian waist, or a place where the wavefront is flat, and w0 is called the beam waist radius. 80 The irradiance distribution of the Gaussian TEM00 beam, namely, 22 2 22 ()==−−2rw//P 2rw , (2.6) Ir Ie0 2 e 60 pw Gaussian Beam Optics where w = w(z) and P is the total power in the beam, is the same at all 40 cross sections of the beam. The invariance of the form of the distribution is a special consequence PERCENT IRRADIANCE 20 = 13.5 of the presumed Gaussian distribution at z 0. If a uniform irradiance distribution had been presumed at z = 0, the pattern at z =∞would have been the familiar Airy disc pattern given by a Bessel function, whereas the 41.5w 4w 0 1.5w pattern at intermediate z values would have been enormously complicated. CONTOUR RADIUS Simultaneously, as R(z) asymptotically approaches z for large z, w(z) asymptotically approaches the value Figure 2.1 Irradiance profile of a Gaussian TEM00 mode lz wz()= (2.7) (intensity) has fallen to 50 percent of its peak, or axial value, as shown pw0 Optical Specifications in figure 2.2. This second definition is also referred to as FWHM, or full 2 width at half maximum. For the remainder of this guide, we will be using where z is presumed to be much larger than pw0 /l so that the 1/e the 1/e2 definition. irradiance contours asymptotically approach a cone of angular radius Diffraction causes light waves to spread transversely as they propagate, () = wz = l . (2.8) and it is therefore impossible to have a perfectly collimated beam. The v zwp 0 spreading of a laser beam is in precise accord with the predictions of pure diffraction theory; aberration is totally insignificant in the present context. Under quite ordinary circumstances, the beam spreading can be so small it can go unnoticed. The following formulas accurately describe beam spread- ing, making it easy to see the capabilities and limitations of laser beams. Even if a Gaussian TEM00 laser-beam wavefront were made perfectly flat 1/e2 diameter 13.5% of peak at some plane, it would quickly acquire curvature and begin spreading in accordance with Material Properties FWHM diameter 50% of peak ⎡ ⎛ 2 ⎞ 2 ⎤ pw0 Rz()=+ z⎢1 ⎜ ⎟ ⎥ (2.4) ⎢ ⎝ ⎠ ⎥ ⎣ lz ⎦ and direction of propagation 12/ ⎡ ⎛ ⎞ 2 ⎤⎤ ⎢ lz ⎥ wz()=+ w0 1 ⎜ ⎟ ⎢ ⎝ 2 ⎠ ⎥ (2.5) ⎣ pw0 ⎦ where z is the distance propagated from the plane where the wavefront 2 is flat, l is the wavelength of light, w0 is the radius of the 1/e irradiance contour at the plane where the wavefront is flat, w(z) is the radius of the 1/e2 contour after the wave has propagated a distance z, and R(z) Optical Coatings is the wavefront radius of curvature after propagating a distance z. R(z) is Figure 2.2 Diameter of a Gaussian beam infinite at z = 0, passes through a minimum at some finite z, and rises again Gaussian Beam Optics 2.3 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:54 PM Page 2.4 Gaussian Beam Optics www.cvimellesgriot.com Fundamental Optics This value is the far-field angular radius (half-angle divergence) of the curvature is a maximum.